Compactification of M(atrix) theory on noncommutative toroidal orbifolds

Abstract

It was shown by A. Connes, M. Douglas and A. Schwarz that noncommutative tori arise naturally in consideration of toroidal compactifications of M(atrix) theory. A similar analysis of toroidal Z_{2} orbifolds leads to the algebra B_{\theta} that can be defined as a crossed product of noncommutative torus and the group Z_{2}. Our paper is devoted to the study of projective modules over B_{\theta} (Z_{2}-equivariant projective modules over a noncommutative torus). We analyze the Morita equivalence (duality) for B_{\theta} algebras working out the two-dimensional case in detail.Comment: 19 pages, Latex; v2: comments clarifying the duality group structure added, section 5 extended, minor improvements all over the tex

Noncommutative field theory

“…The striking feature of this is that it provides a way to define quotients by "bad" group actions, those for which the quotient space M/G is pathological, as is discussed in Connes (1994). This definition of quotient also follows from the standard string theory definition of orbifolds, as discussed in (Douglas, 1999;Konechny and Schwarz, 2000a;Martinec and Moore, 2001).…”

Toroidal orbifolds of Z3 and Z6 symmetries of noncommut

“…For instance, in [14] Konechny and Schwarz used K-theoretical topological invariants (obtained by the author in [18]) to study the structure of projective modules over non-commutative Z 2 orbifolds that admit constant curvature Yang-Mills field as well as obtaining their moduli spaces. And in [13], they study moduli spaces of (equivariant) connections with constant curvature on modules over non-commutative even-dimensional tori and on toroidal orbifolds arising from symmetries of the groups Z 2 and Z 4 (with respective actions from the flip and Fourier transform).…”

Matrix theory compactification on noncommutative T4/Z2

“…7, an A -module is a finitely generated projective module if and only if its corresponding module over A ' ␣ Z 2 is finitely generated projective. As noted in Ref.…”

“…7, there might be other Z 2 actions on the module. Thus we have a solution for ͑6͒, i.e., ⍀ 0 together with U i 's define a projective module over A 'Z 2 .…”